The effects of the spin-orbit interaction in systems with Kondo impurity has been a subject of study in recent decades with more intense discussions in recent years. Much of this interest is stimulated by the importance that the spin-orbit interaction has gained in the recent years in the context of condensed matter physics. In this work we revisit this problem in one-dimensional systems with strong Rashba as well as Dresselhaus spin-orbit interactions. More specifically, we investigate how the spin-orbit interaction modifies the Kondo temperature (TK ) of the system. We begin by describing the system via the Anderson model, taking into account the spin-orbit interactions for the conduction electrons. From this model we derive an effective Kondo-like Hamiltonian that describes not only the conventional Kondo effect, but also accounts for two additional terms. The first therm descri- bes the known Dzyaloshinskii Moriya interaction, while the second one describes electron scattering processes similar to those described by Elliot-Yafet. To determine the effects of the spin-orbit in- teraction on TK , we analyze the effective Hamiltonian via renormalization group. Our study shows that, due to a renormalization of the Kondo coupling, there is an increase of the Kondo temperature with increasing spin-orbit interaction. This occurs even at the particle-hole symmetry point, differing from recent results in the literature. The additional terms that appear in the Hamiltonian also contribute to the increase of TK away from the particle-hole symmetry. Moreover, the combination of the Dzyaloshinskii-Moriya and Elliot-Yafet mechanisms produce a dependency of the Kondo temperature on the spin-orbit coupling that is asymmetric regarding displacements of the impurity level position around the particle-hole symmetric point.